The Bethe Ansatz

Models and eigenstates: the Coordinate Bethe Ansatz
The Lieb-Liniger gas
Bogoliubov theory

Bogoliubov theoryc.l.b

It is possible to effectively resum terms in perturbation theory by following Bogoliubov's logic for quasi-degenerate gases. This begins by assuming that the lowest momentum mode is macroscopically occupied, namely that

\hat{N} = {\hat{N}}_{0} + \frac{1}{L} \sum_{k \neq 0} ψ_{k}^{†} ψ_{k}, \frac{1}{L} ⟨ ψ_{k = 0}^{†} ψ_{k = 0} ⟩ \equiv N_{0} = O (N) .

Keeping only leading terms and performing a Bogoliubov transformation then gives the simplified Hamiltonian

\begin{matrix} (l.hb) & H_{L L}^{B o g} = E_{0} + \frac{1}{L} \sum_{k \neq 0} ε^{B o g} (k) {\tilde{ψ}}_{k}^{†} {\tilde{ψ}}_{k} \end{matrix}

where $ε^{B o g}$ is the well-known Bogoliubov dispersion relation

\begin{matrix} (l.eb) & ε^{B o g} (k) = {[k^{4} + 4 c n k^{2}]}^{1 / 2}, n = N / L \end{matrix}

and where the ground state energy is

\begin{matrix} (l.e0b) & e_{0}^{B o g} \equiv \frac{E_{0}}{L} = c n^{2} - \frac{4}{3 π} (c n)^{3 / 2} = n^{3} γ (1 - \frac{4}{3 π} \sqrt{γ}) \end{matrix}

in terms of the effective interaction parameter $γ = c / n$ .

Derivation

Following Bogoliubov's logic, we assume that the lowest momentum mode is macroscopically occupied, namely that

\hat{N} = {\hat{N}}_{0} + \frac{1}{L} \sum_{k \neq 0} ψ_{k}^{†} ψ_{k}, \frac{1}{L} ⟨ ψ_{k = 0}^{†} ψ_{k = 0} ⟩ \equiv N_{0} = O (N) .

In the Lieb-Liniger Hamiltonian represented in Fourier space l.hf, keeping only the leading terms involving at least two entries of the zero-momentum fields gives

H_{L L} ≃ \frac{c}{L^{3}} Ψ_{0}^{†} Ψ_{0}^{†} Ψ_{0} Ψ_{0} + \frac{1}{L} \sum_{k} k^{2} Ψ_{k}^{†} Ψ_{k} + \frac{2 c}{L^{3}} \sum_{k > 0} [4 Ψ_{0}^{†} Ψ_{k}^{†} Ψ_{0} Ψ_{k} + (Ψ_{0}^{†})^{2} Ψ_{k} Ψ_{- k} + Ψ_{k}^{†} Ψ_{- k}^{†} Ψ_{0}^{2}] .

Simple algebra then gives

H_{L L}^{B o g} = \frac{c N^{2}}{L} - \sum_{k > 0} (k^{2} + 2 c n) + \frac{1}{L} \sum_{k > 0} (k^{2} + 2 c n) (\begin{array}{cc} ψ_{k}^{†} & ψ_{- k} \end{array}) (\begin{array}{cc} 1 & γ_{k} \\ γ_{k} & 1 \end{array}) (\begin{matrix} ψ_{k} \\ ψ_{- k}^{†} \end{matrix})

where $n = \frac{N}{L}$ and

γ_{k} \equiv \frac{2 c n}{k^{2} + 2 c n} .

This can be diagonalized by a Bogoliubov transformation

(\begin{matrix} ψ_{k} \\ ψ_{- k}^{†} \end{matrix}) = (\begin{array}{cc} \cosh θ_{k} & \sinh θ_{k} \\ \sinh θ_{k} & \cosh θ_{k} \end{array}) (\begin{matrix} {\tilde{ψ}}_{k} \\ {\tilde{ψ}}_{- k}^{†} \end{matrix}), γ_{k} = \tanh 2 θ_{k} .

The diagonalized quadratic form in the Hamiltonian then becomes $\frac{1}{\cosh 2 θ_{k}} 1 = {[1 - γ_{k}^{2}]}^{1 / 2}$ . The excitation energy $(k^{2} + 2 c n) (1 - γ_{k}^{2})$ then gives the well-known Bogoliubov dispersion relation

ε_{B o g} (k) = {[k^{4} + 4 c n k^{2}]}^{1 / 2} .

The Hamiltonian itself simplifies to

H_{L L}^{B o g} = E_{0} + \frac{1}{L} \sum_{k \neq 0} ε_{B o g} (k) {\tilde{ψ}}_{k}^{†} {\tilde{ψ}}_{k}

where the ground state energy is

\frac{E_{0}}{L} = c n^{2} + \int_{0}^{\infty} \frac{d k}{2 π} ([k^{4} + 4 c n k^{2}]^{1 / 2} - k^{2} - 2 c n) = c n^{2} - \frac{4}{3 π} (c n)^{3 / 2} = n^{3} γ (1 - \frac{4}{3 π} \sqrt{γ})

in terms of the effective interaction parameter $γ = c / n$ .

The Bogoliubov approach should provide an accurate approximation of the Lieb-Liniger model in the limit of small interactions. Note the the ground state energy has non-algebraic corrections in terms of the interaction parameter, reflecting the fact that naive perturbation theory around the noninteracting point fails. Bogoliubov theory cannot be accurate for large interactions, in any case certainly not for $γ > (3 π / 4)^{2} \approx 5.55$ , where the prediction for the ground state energy becomes negative. A slightly more refined limit is that since the ground-state energy must be a monotonically increasing function of the interaction (this being a simple consequence of the Hellman-Feynman theorem $\frac{d E_{0}}{d c} = ⟨ \frac{d H}{d c} ⟩_{0}$ ), Bogoliubov theory cannot be a valid approximation for the Lieb-Liniger model when $γ > (π / 2)^{2} \approx 2.47$ . Of course, how accurate Bogoliubov theory is depends very much on which question is being asked. We will comment on its validity for various quantities at relevant points in our study of the physics of the Lieb-Liniger model.

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Author: Jean-Sébastien Caux

Created: 2024-01-18 Thu 14:24

Bogoliubov theory c.l.b

Bogoliubov theoryc.l.b